Non-definite matrices? HELP!

So I am trying to determine if the difference of two matrices is positive(negative) semi-definite. The matrices, we will call them and , are partitioned as

and.

My problem is that I wish to determine the definiteness of the matrix or/and . These matrices are given by
and

We know from simple linear algebra that a matrix is positive(negative) semi-definite if and only if all of its principle minors are non-negative(non-positive). With this I have two questions: 1) Does the same hold true for block matrices? 2) If this does hold true the second principle minor, in block form, for both matrices is , what does this mean exactly. Does this mean that the matrices are non-definite?

Re: Non-definite matrices? HELP!

Ok, I found out that I can use the principle minor characterization of block matrices. However, there is still a problem. While the determinant of the first principle minor is zero. The determinant of the entire matrix(in block form) is .

Re: Non-definite matrices? HELP!

Originally Posted by frazdt

So I am trying to determine if the difference of two matrices is positive(negative) semi-definite. The matrices, we will call them and , are partitioned as

and

Something strange is going on here. For the (1,2)-element of to be defined, the matrices and must be the same size (same number of rows and columns). Looking at the other elements of , you see that and must all be the same size. That implies that they are all square matrices of the same size.

In any case, the matrix is only defined if is a square matrix. In general, there is nothing you can say about the positivity or otherwise of It could be positive definite, negative definite or (more probably) neither.